Compact difference method for solving the fractional reaction–subdiffusion equation with Neumann boundary value condition

In this paper, we derive a high-order compact finite difference scheme for solving the reaction–subdiffusion equation with Neumann boundary value condition. The L1 method is used to approximate the temporal Caputo derivative, and the compact difference operator is applied for spatial discretization. We prove that the compact finite difference method is unconditionally stable and convergent with order O(τ^2?α+h⁴) in L₂ norm, where τ, α, and h are the temporal step size, the order of time fractional derivative and the spatial step size, respectively. Finally, some numerical experiments are carried out to show the effectiveness of the proposed difference scheme.     

Sextic spline solution of variable coefficient fourth-order parabolic equations

We report new three-level implicit methods of O(k ²+h ⁴) and O(k ⁴+h ⁴) for the numerical solution of fourth-order non-homogeneous parabolic partial differential equation with variable coefficients. We use sextic spline in space and finite difference in time directions. Sextic spline function relations are derived by using off-step points. The linear stability of the presented method is investigated. We solve test problems numerically to validate the derived methods. Numerical comparison with other existence methods shows the superiority of our presented scheme.     

A Refined Finite Element Convergence Theory for Highly Indefinite Helmholtz Problems

It is well known that standard h-version finite element discretisations using lowest order elements for Helmholtz' equation suffer from the following stability condition: ``The mesh width h of the finite element mesh has to satisfy k <sup>2</sup> h≲1', where k denotes the wave number. This condition rules out the reliable numerical solution of Helmholtz equation in three dimensions for large wave numbers k≳50. In our paper, we will present a refined finite element theory for highly indefinite Helmholtz problems where the stability of the discretisation can be checked through an ``almost invariance' condition. As an application, we will consider a one-dimensional finite element space for the Helmholtz equation and apply our theory to prove stability under the weakened condition hk≲1 and optimal convergence estimates. Dedicated to Prof. Dr. Ivo Babuška on the occasion of his 80th birthday.     

New local discontinuous Galerkin method for a fractional time diffusion wave equation

ABSTRACT

This paper is devoted to the study of a new discontinuous finite element idea for the time fractional diffusion-wave equation defined in bounded domain. The time fractional derivatives are described in the Caputo's sense. By applying the sine transform on the time fractional diffusion-wave equation, we make the equation depend on time. Then we use definition of Caputo's derivative and by defining l-degree discontinuous finite element with interpolated coefficients we solve the mentioned equation. Error estimate, existence and uniqueness are proved. Finally, the theoretical results are tested by some numerical examples.     

Finite difference method for time-space-fractional Schrödinger equation

In this paper, an implicit finite difference scheme for the nonlinear time-space-fractional Schrödinger equation is presented. It is shown that the implicit scheme is unconditionally stable with experimental convergence order of O(τ^2?α+h²), where τ and h are time and space stepsizes, respectively, and α (0<α<1) is the fractional-order in time. In order to reduce the computational cost, the explicit–implicit scheme is proposed such that the nonlinear term is easily treated. Meanwhile, the implicit finite difference scheme for the coupled time-space-fractional Schrödinger system is also presented, which is unconditionally stable too. Numerical examples are given to support the theoretical analysis.     

Numerical approximation of nonlinear fractional differential equations with subdiffusion and superdiffusion

In this paper, we study the time-space fractional order (fractional for simplicity) nonlinear subdiffusion and superdiffusion equations, which can relate the matter flux vector to concentration gradient in the general sense, describing, for example, the phenomena of anomalous diffusion, fractional Brownian motion, and so on. The semi-discrete and fully discrete numerical approximations are both analyzed, where the Galerkin finite element method for the space Riemann-Liouville fractional derivative with order 1+β∈[1,2] and the finite difference scheme for the time Caputo derivative with order α∈(0,1) (for subdiffusion) and (1,2) (for superdiffusion) are analyzed, respectively. Results on the existence and uniqueness of the weak solutions, the numerical stability, and the error estimates are presented. Numerical examples are included to confirm the theoretical analysis. During our simulations, an interesting diffusion phenomenon of particles is observed, that is, on average, the diffusion velocity for 0<α<1 is slower than that for α=1, but the diffusion velocity for 1<α<2 is faster than that for α=1. For the spatial diffusion, we have a similar observation.     

Finite central difference/finite element approximations for parabolic integro-differential equations

We study the numerical solution of an initial-boundary value problem for parabolic integro-differential equation with a weakly singular kernel. The main purpose of this paper is to construct and analyze stable and high order scheme to efficiently solve the integro-differential equation. The equation is discretized in time by the finite central difference and in space by the finite element method. We prove that the full discretization is unconditionally stable and the numerical solution converges to the exact one with order O(Δt ² + h ^l ). A numerical example demonstrates the theoretical results.     

Numerical solution for a class of space-time fractional equation by the piecewise reproducing kernel method

ABSTRACT

Due to the non-locality of fractional derivative, the analytical solution and good approximate solution of fractional partial differential equations are usually difficult to get. Reproducing kernel space is a perfect space in studying this type of equations, however the numerical results of equations by using the traditional reproducing kernel method (RKM) isn't very good. Based on this problem, we present the piecewise technique in the reproducing kernel space to solve this type of equations. The focus of this paper is to verify the stability and high accuracy of the present method by comparing the absolute error with traditional RKM and study the effect on absolute error for different values of α. Furthermore, we can study the distribution of entire space at a particular time period. Three numerical experiments are provided to verify the efficiency and stability of the proposed method. Meanwhile, it is tested by experiments that the change of the value of α has little effect on its accuracy.     

A New Nonsymmetric Discontinuous Galerkin Method for Time Dependent Convection Diffusion Equations

We propose a discontinuous Galerkin finite element method for convection diffusion equations that involves a new methodology handling the diffusion term. Test function derivative numerical flux term is introduced in the scheme formulation to balance the solution derivative numerical flux term. The scheme has a nonsymmetric structure. For general nonlinear diffusion equations, nonlinear stability of the numerical solution is obtained. Optimal kth order error estimate under energy norm is proved for linear diffusion problems with piecewise P ^k polynomial approximations. Numerical examples under one-dimensional and two-dimensional settings are carried out. Optimal (k+1)th order of accuracy with P ^k polynomial approximations is obtained on uniform and nonuniform meshes. Compared to the Baumann-Oden method and the NIPG method, the optimal convergence is recovered for even order P ^k polynomial approximations.     

Spectral regularization method for the time fractional inverse advection-dispersion equation

In this paper, we consider the time fractional inverse advection-dispersion problem (TFIADP) in a quarter plane. The solute concentration and dispersion flux are sought from a measured concentration history at a fixed location inside the body. Such problem is obtained from the classical advection-dispersion equation by replacing the first-order time derivative by the Caputo fractional derivative of order α(0 < α < 1). We show that the TFIADP is severely ill-posed and further apply a spectral regularization method to solve it based on the solution given by the Fourier method. Convergence estimates are presented under a priori bound assumptions for the exact solution. Finally, numerical examples are given to show that the proposed numerical method is effective.     

Non-polynomial cubic spline methods for the solution of parabolic equations

Second-order parabolic partial differential equations are solved by using a new three level method based on non-polynomial cubic spline in the space direction and finite difference in the time direction. Stability analysis of the method has been carried out and we have shown that our method is unconditionally stable. It has been shown that by suitably choosing the parameters most of the previous known methods for homogeneous and non-homogeneous cases can be obtained from our method. We also obtain a new high accuracy scheme of O(k ⁴+h ⁴). Numerical examples are given to illustrate the applicability and efficiency of the new method.     

The WKB Local Discontinuous Galerkin Method for the Simulation of Schrödinger Equation in a Resonant Tunneling Diode

In this paper, we develop a multiscale local discontinuous Galerkin (LDG) method to simulate the one-dimensional stationary Schrödinger-Poisson problem. The stationary Schrödinger equation is discretized by the WKB local discontinuous Galerkin (WKB-LDG) method, and the Poisson potential equation is discretized by the minimal dissipation LDG (MD-LDG) method. The WKB-LDG method we propose provides a significant reduction of both the computational cost and memory in solving the Schrödinger equation. Compared with traditional continuous finite element Galerkin methodology, the WKB-LDG method has the advantages of the DG methods including their flexibility in h-p adaptivity and allowance of complete discontinuity at element interfaces. Although not addressed in this paper, a major advantage of the WKB-LDG method is its feasibility for two-dimensional devices.     

High-accuracy cubic spline alternating group explicit methods for 1D quasi-linear parabolic equations†

In this article, we study the application of the alternating group explicit (AGE) and Newton-AGE iterative methods to a two-level implicit cubic spline formula of O(k ²+kh ²+h ⁴) for the solution of 1D quasi-linear parabolic equation u _xx =φ (x, t, u, u _x , u _t ), 0<x<1, t>0 subject to appropriate initial and natural boundary conditions prescribed, where k>0 and h>0 are mesh sizes in t- and x-directions, respectively. The proposed cubic spline methods require 3-spatial grid points and are applicable to problems in both rectangular and polar coordinates. The convergence analysis at advanced time level is briefly discussed. The proposed methods are then compared with the corresponding successive over relaxation (SOR) and Newton-SOR iterative methods both in terms of accuracy and performance.     

Local Discontinuous Galerkin Method for Surface Diffusion and Willmore Flow of Graphs

In this paper, we develop a local discontinuous Galerkin (LDG) finite element method for surface diffusion and Willmore flow of graphs. We prove L ² stability for the equation of surface diffusion of graphs and energy stability for the equation of Willmore flow of graphs. We provide numerical simulation results for different types of solutions of these two types of the equations to illustrate the accuracy and capability of the LDG method.     

Tension spline approach for the numerical solution of nonlinear Klein-Gordon equation

The nonlinear Klein-Gordon equation describes a variety of physical phenomena such as dislocations, ferroelectric and ferromagnetic domain walls, DNA dynamics, and Josephson junctions. We derive approximate expressions for the dispersion relation of the nonlinear Klein-Gordon equation in the case of strong nonlinearities using a method based on the tension spline function and finite difference approximations. The resulting spline difference schemes are analyzed for local truncation error, stability and convergence. It has been shown that by suitably choosing the parameters, we can obtain two schemes of O(k²+k²h²+h²) and O(k²+k²h²+h⁴). In the end, some numerical examples are provided to demonstrate the effectiveness of the proposed schemes.     

A new unconditionally stable ADI compact scheme for the two-space-dimensional linear hyperbolic equation

We formulate a new alternating direction implicit compact scheme of O(τ²+h ⁴) for the linear hyperbolic equation u _tt +2α u _t +β² u=u _xx +u _yy +f(x, y, t), 0<x, y<1, 0<t≤T, subject to appropriate initial and Dirichlet boundary conditions, where α>0 and β≥0 are real numbers. In this article, we show the method is unconditionally stable by the Von Neumann method. At last, numerical demonstrations are given to illustrate our result.     

The new numerical method for solving the system of two-dimensional Burgers’ equations

In this paper, the system of two-dimensional Burgers’ equations are solved by local discontinuous Galerkin (LDG) finite element method. The new method is based on the two-dimensional Hopf–Cole transformations, which transform the system of two-dimensional Burgers’ equations into a linear heat equation. Then the linear heat equation is solved by the LDG finite element method. The numerical solution of the heat equation is used to derive the numerical solutions of Burgers’ equations directly. Such a LDG method can also be used to find the numerical solution of the two-dimensional Burgers’ equation by rewriting Burgers’ equation as a system of the two-dimensional Burgers’ equations. Three numerical examples are used to demonstrate the efficiency and accuracy of the method.     

An analytic algorithm for the space–time fractional advection–dispersion equation

Fractional advection–dispersion equation (FADE) is a generalization of the classical ADE in which the first order time derivative and first and second order space derivatives are replaced by Caputo derivatives of orders 0<α?1, 0<β?1 and 1<γ?2, respectively. We use Caputo definition to avoid (i) mass balance error, (ii) hyper-singular improper integral, (iii) non-zero derivative of constant, and (iv) fractional derivative involved in the initial condition which is often ill-defined. We present an analytic algorithm to solve FADE based on homotopy analysis method (HAM) which has the advantage of controlling the region and rate of convergence of the solution series via the auxiliary parameter ? over the variational iteration method (VIM) and homotopy perturbation method (HPM). We find that the proposed method converges to the numerical/exact solution of the ADE as the fractional orders α, β, γ tend to their integral values. Numerical examples are given to illustrate the proposed algorithm. Example 5 describes the intermediate process between advection and dispersion via Caputo fractional derivative.     

Numerical Simulation for Porous Medium Equation by Local Discontinuous Galerkin Finite Element Method

In this paper we will consider the simulation of the local discontinuous Galerkin (LDG) finite element method for the porous medium equation (PME), where we present an additional nonnegativity preserving limiter to satisfy the physical nature of the PME. We also prove for the discontinuous ℙ₀ finite element that the average in each cell of the LDG solution for the PME maintains nonnegativity if the initial solution is nonnegative within some restriction for the flux’s parameter. Finally, numerical results are given to show the advantage of the LDG method for the simulation of the PME, in its capability to capture accurately sharp interfaces without oscillation. The research of Q. Zhang is supported by CNNSF grant 10301016.